Robert, if your siren needs an absolute minimum of 12 V, it's likely not going to work anyway with a 12V battery, as the battery could get under 12V fairly quickly when being discharged depending on the load and it's capacity. I think you need a proper spec for what the circuit produces i.e. "Must operate from a 12V sealed lead acid battery and/or an external power supply between 12V and 16V and must produce 12 V regulated to +/- 0.5 V"... that sort of thing. Further, you need to specify with tolerances, the current needed and for how long, and under what conditions. If you plan to mount it outside in a garage or something, the temperature range will be important too.

The circuit shown is way too complicated for what it does - effectively switching the load to one supply or another. You would not need 2 FETs for each leg - power VFETS have a saturated resistance of a few milliohms depending on the device chosen. At 5A you wouldn't even need a heat sink. And how does the battery get charged? You might be able to connect the battery charger to the battery and power from the battery at the same time (like a car) depending on your spec. Or you might need to design either a switching supply, or a linear regulator that can work with a very small volt drop - again depending on that spec.

It's a fun problem.